A laser emits 1.42 1018 photons per second in a beam of light that has a diameter of 1.82 mm and a wavelength of 524.0 nm. Determine each of the following for the electromagnetic wave that constitutes the beam.
(a) the average electric field strength
(b) the average magnetic field strength
S=Energy per unit time/Area = Nhf/A= =4Nhc/πD²λ=
=4•1.42•10¹⁸•6.63•10⁻³⁴•3•10⁸/π•(1.82•10⁻³)²•524•10⁻⁹=
=2.07•10⁵ W/m²
Electric field
E= sqrt{S/(ε₀c)}=
= sqrt{2.07•10⁵/8.85•10⁻²•3•10⁸}=8834 N/C
Magnetic field
B=E/c =8834/3•10⁸ =2.94•10⁻⁵T
A laser emits a beam of light whose photons all have the same frequency. When the beam strikes the surface of a metal, photo-electrons are ejected from the surface. What happens if the laser emits twice the number of photons per second?
To determine the average electric and magnetic field strengths for the electromagnetic wave, we can use the following formulas:
(a) Average Electric Field Strength:
The average electric field strength (E_avg) can be calculated using the formula:
E_avg = c * sqrt(2 * P / (A * ε₀))
where:
- E_avg is the average electric field strength
- c is the speed of light in vacuum (approximately 3.00 x 10^8 m/s)
- P is the power of the laser (number of photons emitted per second)
- A is the cross-sectional area of the beam
- ε₀ is the permittivity of free space (approximately 8.85 x 10^-12 F/m)
Given values:
P = 1.42 x 10^18 photons per second
A = π * (d/2)^2, where d is the diameter of the beam (1.82 mm)
ε₀ = 8.85 x 10^-12 F/m
Substituting the values into the formula, we can calculate E_avg.
(b) Average Magnetic Field Strength:
The average magnetic field strength (B_avg) can be calculated using the formula:
B_avg = E_avg / c
where:
- B_avg is the average magnetic field strength
- E_avg is the average electric field strength
- c is the speed of light in vacuum (approximately 3.00 x 10^8 m/s)
Given that we have already calculated E_avg, we can now calculate B_avg.
Now, let's plug in the values and calculate step by step:
(a) Average Electric Field Strength:
Step 1: Calculate the cross-sectional area of the beam
d = 1.82 mm = 1.82 x 10^-3 m (converting millimeters to meters)
A = π * (1.82 x 10^-3 / 2)^2
A = π * (9.13 x 10^-4)^2
A ≈ 2.62 x 10^-6 m²
Step 2: Calculate E_avg
E_avg = c * sqrt(2 * P / (A * ε₀))
E_avg = (3.00 x 10^8) * sqrt(2 * (1.42 x 10^18) / (2.62 x 10^-6 * 8.85 x 10^-12))
Now, let's calculate the value of E_avg.
(b) Average Magnetic Field Strength:
Step 3: Calculate B_avg
B_avg = E_avg / c
Now, let's calculate the value of B_avg.
Please provide me with a moment to perform the calculations.
To determine the average electric field strength and average magnetic field strength of the electromagnetic wave, we can use the following formulas:
(a) Average electric field strength (E):
E = c * sqrt(2 * P / (ε₀ * A))
(b) Average magnetic field strength (B):
B = sqrt((2 * P) / (c * μ₀ * A))
Where:
- c is the speed of light in a vacuum (3.00 x 10^8 m/s)
- P is the power of the laser (1.42 x 10^18 photons/s)
- ε₀ is the vacuum permittivity (8.85 x 10^-12 F/m)
- A is the cross-sectional area of the beam (π * r^2, where r is the radius of the beam)
- μ₀ is the vacuum permeability (4π x 10^-7 Tm/A)
Now let's calculate each of the values:
(a) Average electric field strength (E):
First, we need to calculate the radius of the beam (r) from the given diameter:
diameter = 2 * r
1.82 mm = 2 * r
r = 0.91 mm = 0.00091 m
Next, we calculate the cross-sectional area (A):
A = π * (0.00091 m)^2 = 2.61 x 10^-6 m^2
Finally, we can substitute these values into the formula to find E:
E = (3.00 x 10^8 m/s) * sqrt(2 * (1.42 x 10^18) / ((8.85 x 10^-12 F/m) * (2.61 x 10^-6 m^2)))
(b) Average magnetic field strength (B):
We can use the same values for the radius (r) and area (A) as in part (a).
Now we calculate B using the formula:
B = sqrt((2 * (1.42 x 10^18)) / ((3.00 x 10^8 m/s) * (4π x 10^-7 Tm/A) * (2.61 x 10^-6 m^2)))
By substituting the values into the formulas, you should be able to calculate the average electric field strength (E) and average magnetic field strength (B) for the given beam of light.